Question

Abstract Algebra help

Answer 1

let ,\[G=<a>\] ,|a|=12 and \[K=<a ^{3}>\].Find the distinct cosets of K in G

Answer 2

how am i going to find a

Answer 3

If I understood correctly, G is cyclic, generated by a.. and it has 12 elements total, right?

Answer 4

ok i will keep on waiting

Answer 5

i think the elements will be
G= \[\left\{ e,a ^{2},a ^{3},a ^{4},a ^{5},a ^{6},a ^{7},a ^{8},a ^{9},a ^{10},a ^{11} \right\}\]

Answer 6

i omit " a" that should be the second element

Answer 7

\[\large G=\left\{ e, \ a \ ,a ^{2},a ^{3},a ^{4},a ^{5},a ^{6},a ^{7},a ^{8},a ^{9},a ^{10},a ^{11} \right\}\]

Answer 8

Seems you're back :)
\[\large K=<a^3> = \left\{ e,a^3,a^6,a^9\right\}\]

Answer 9

yes i see it

Answer 10

it means i should multiply k by G

Answer 11

not sure

Answer 12

so i think this shows that [G:K]=3 BY lagrange but i don't know "should i find them one by one by multiplying k by any element in G"

Answer 13

Wait, what's your current question?

Answer 14

Find the distinct cosets

Answer 15

Okay, so, the cosets of G:K are
\[\huge\left\{gK|g\in G\right\}\]

Answer 16

yep

Answer 17

Well, go ahead, pick an element of G, any one of them, except those that are already in K :)

Answer 18

ok it seems i was doing the correct things yesterday night.

i found that \[\left\{ k,ak,a ^{2}k \right\}\] where \[ak=\left\{ a,a ^{4},a ^{7},a^{10} \right\}\] ,\[a ^{2}k= \left\{ a ^{2},a ^{5},a ^{8},a ^{11} \right\} \] and K

Answer 19

That's correct. Indeed :)

Answer 20

by multiplying all element in G BY K

Answer 21

Yes, well, some elements of G would give the same coset of K, but you already found that out, right? :)

Answer 22

yes i found then on my rough work :)

Answer 23

Nice :)

Answer 24

i have the question under normal subgroup

Answer 25

Okay, let's do this :)

Answer 26

Let H\[\le G\] and ,SHOW THAT H intersection N \[\left( H n N \right) \Delta H\]